Recognizing hyperelliptic graphs in polynomial time

Based on analogies between algebraic curves and graphs, Baker and Norine introduced divisorial gonality, a graph parameter for multigraphs related to treewidth, multigraph algorithms and number theory. Various equivalent definitions of the gonality of an algebraic curve translate to differentnotions of gonality for graphs, called stable gonalityand stable divisorial gonality. We consider so-called hyperelliptic graphs(multigraphs of gonality 2, in any meaning of graph gonality) and provide a safe and complete set of reduction rules for such multigraphs. This results in an algorithm to recognize hyperelliptic graphs in time O(m + n logn), where nis the number of vertices and mthe number of edges of the multigraph. A corollary is that we can decide with the same runtime whether a two-edge-connected graph Gadmits an involution ssuch that the quotient G/<sigma > is a tree. (C) 2020 The Authors. Published by Elsevier B.V.